Solve for $x$ : $ 4|x + 7| - 3 = 2|x + 7| + 2 $
Solution: Subtract $ {2|x + 7|} $ from both sides: $ \begin{eqnarray} 4|x + 7| - 3 &=& 2|x + 7| + 2 \\ \\ { - 2|x + 7|} && { - 2|x + 7|} \\ \\ 2|x + 7| - 3 &=& 2 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 2|x + 7| - 3 &=& 2 \\ \\ { + 3} &=& { + 3} \\ \\ 2|x + 7| &=& 5 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x + 7|} {{2}} = \dfrac{5} {{2}} $ Simplify: $ |x + 7| = \dfrac{5}{2}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -\dfrac{5}{2} $ or $ x + 7 = \dfrac{5}{2} $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -\dfrac{5}{2} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -\dfrac{5}{2} \\ \\ {- 7} && {- 7} \\ \\ x &=& -\dfrac{5}{2} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $2$ $ x = - \dfrac{5}{2} {- \dfrac{14}{2}} $ $ x = -\dfrac{19}{2} $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = \dfrac{5}{2} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& \dfrac{5}{2} \\ \\ {- 7} && {- 7} \\ \\ x &=& \dfrac{5}{2} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $2$ $ x = \dfrac{5}{2} {- \dfrac{14}{2}} $ $ x = -\dfrac{9}{2} $ Thus, the correct answer is $x = -\dfrac{19}{2} $ or $x = -\dfrac{9}{2} $.